The way I understand this, you have a three sided coop with wire on the fourth side. It is inside an enclosed run. I'm assuming that since you want an automatic pop door, you will not be there to open and close it every morning and night.
Whether or not you need a pop door depends on your confidence in the run. It is possible to build a predatro proof run. The bigger it is, the more expensive it gets. It also takes more dilligence to make sure you don't have a weak spot where something can climb, dig, or squeeze through, especially if the run gets to be any size. But it is possible.
The other thing to consider is the predators you are protecting against. It is real hard to keep snakes, rats, and some weasels out. Raccoons and foxes are easier. But any of them can exploit a weakness in your defenses.
In your climate, I really don't know of any reason other than predators to worry about having a pop door or an open side. You can use a pop door and use hardwire cloth on the open side to make it real predator proof if you wish. It is just up to your confidence in how predator proof your run is.
One of the automatic pop door vendors is an advertiser on this site. I don't know how good that particular door is, but you might want to contact a moderator for contact information.